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From Wikipedia

Polar coordinate system

In mathematics, the polar coordinate system is a two-dimensionalcoordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction.

The fixed point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole with the fixed direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is the angular coordinate, polar angle, or azimuth.


The concepts of angle and radius were already used by ancient peoples of the 1st millennium BCE. The Greek astronomer and astrologerHipparchus (190-120 BCE) created a table of chord functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions. In On Spirals,Archimedes describes the Archimedean spiral, a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system.

From the 8th century CE onward, Muslim astronomers developed methods for approximating and calculating the direction to Makkah (qibla)—and its distance—from any location on the Earth. From the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately. The calculation is essentially the conversion of the equatorial polar coordinates of Mecca (i.e. its longitude and latitude) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the great circle through the given location and the Earth's poles, and whose polar axis is the line through the location and its antipodal point.

The Persian geographer, Abū Rayh�n Bīrūnī (973-1048), developed ideas which are seen as an anticipation of the polar coordinate system. Around 1025 CE, he was the first to describe a polar equi-azimuthal equidistant projection of the celestial sphere.

There are various accounts of the introduction of polar coordinates as part of a formal coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidge's Origin of Polar Coordinates.Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653. Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs.

In Method of Fluxions(written 1671, published 1736), SirIsaac Newton examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems. In the journal Acta Eruditorum(1691),Jacob Bernoulli used a system with a point on a line, called the pole and polar axis respectively. Coordinates were specified by the distance from the pole and the angle from the polar axis. Bernoulli's work extended to finding the radius of curvature of curves expressed in these coordinates.

The actual term polar coordinates has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. The term appeared in English in George Peacock's 1816 translation of Lacroix's Differential and Integral Calculus. Alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them.

Common conventions

The radial coordinate is often denoted by r, and the angular coordinate by θ or t.

Angles in polar notation are generally expressed in either degrees or open source program that can generate two dimensional plots of mathematical functions and data sets.


Graph supports entering functions on regular cartesian as well as parametric and polar form. Functions can be traced and at a given coordinate the function value and the two first derivatives is displayed. Graph is available in 23 different languages. A linux version isn't planned, yet the program works well using WINE.

From Yahoo Answers

Question:So I'm a bizarre nerd and I want to transcribe my handwritten math notes to a Word document... I've done searches and found lots of graphing calculators online, even some that purport to do polar coordinates, but I guess I'm not advanced enough to figure them out--graphing calculators were just being invented when I first took higher math and this is my second time around! Right now all I want to do is type in a set of polar coordinates given in examples by my professor and have a graph spit back at me that I can screen capture, crop, and put into place in my notes. I don't need the calculator to do anything fancy like loops and circles yet, but I probably will... maybe after today's class! Best answer will go to the person who either lists a site which is self-explanatory or lists a site with instructions for using the calculator. PLEASE include a description with your answer, not just a link. Thanks!

Answers:I know that you can with maple. You can download a trial version of it, but the non-trial is not free. It is not the easiest version to learn, but it is compatible with word (can copy and paste into word easily). Also if you already have a TI-83 you could use the cord to hook into the computer. I don't know if you can put the graphs into word, but it is worth a try.

Question:Note: I am using t instead of theta For the following problem, we are asked to find the area that the curve encloses: r = 3[1 + Cos(t)] I know that the formula for calculating area in polar coordinates is r^2 dt. I looked at the graph on my calculator and I believe I am correct, but I just want to make sure that I am: is the lower limit of integration 0, and the upper limit of integration 2pi? Or equivalently, is the lower limit of integration 0 and the upper limit of integration pi if I decide to add in a 2 outside of the integral and have it cancel out with the (from r^2 dt)?

Answers:I agree with what you are doing, but do not follow your reason. Here is how I explain it: As t goes from 0 to pi the end of radius r draws out a curve above the horizontal axis. Because the curve does not intersect itself, and because it lies entirely above the horizontal axis (endpoints on the axis), it has area bounded above by the curve and below by the horizontal axis. Call that area A-up. As t goes from pi to 2 pi, the mirror image of the curve is drawn entirely below the horizontal axis, and that part has of the curve has area bounded by the lower curve and the horizontal axis, Alower. Because of the mirror image symmetry, you can calculate the total area either by a full integration over 0 to 2 pi, Afull, or you can calculate the area of the upper half and double it, Afull = 2 Aup ( = 2 Alower). Area- full = Integral of r(t) from 0 to 2 pi Area-up = Integral of r(t) from 0 to pi Area-full = 2 * Area-up I do not believe saying the 2 above is canceling out the 1/2 in the expression 1/2 r^2 dt. In effect one cancels the other, but that is just a coincidence, and not a reason for the method. The reason is the symmetry. If a function r(t) did not have symmetry above and below the horizontal axis, you could not do this. (Example: r = 3[1+cos(t/3))

Question:Hey all I just got a ti-83 for my trigonometry class and Im learning about polar coordinates. I'm not sure how to use the calculator to good yet so could some one tell me how to graph some coordinates such as (r,0) = (2, 3pi/2) when I get to the place to add my coordinates I see this: \r1 = \r2 = \r3 = etc My question is where do I put the r coordinate and where do I put the o or theta coordinate? Thanks!

Answers:the input is r = f( ) It is for plotting functions, not individual points, you could enter r1 = 1 + 2 to get an Archimedean spiral

Question:How would you create a 3d graph with polar coordinates?

Answers:You need to use either cylindrical coordinates or spherical coordinates. Both of them start with polar coordinates and extend them in different ways to 3d.

From Youtube

Polar Coordinates - Basic Graphing :Polar Coordinates - Basic Graphing. In this video I show some basic graphing techniques. For more free math videos, visit UT math tutor, austinmathtutoring, austinmathtutor, austin math tutor, austin algebra tutor, austin calculus tutor

Rectangular, Parametric, and Polar Coordinate Systems This activity helps students understand the differences when working in different coordinate systems. This activity focuses mainly on graphing circles in these systems, but also touches on other ideas through exploration. Students will summarize their findings and judge which coordinate systems are more efficient for graphing different circles. This activity was designed to be completed using the ClassPad 330 graphing calculator.